They have a better chance of finding Jimmy Hoffa in their digs.
It is buried in a chamber that was made for it by King Solomon. It was taken from the Holy of Holies (where it rested on top of the Foundation Stone) at the orders of King Josiah (”Yoshiyahu” in transliterated Hebrew). This was done NOT in haste, but at a time when the people of Judah (and, thus, the Temple Mount and the Ark) were safe. He was King from 641 BCE through 609 BCE, with his reign ending more than 20 years before the destruction of the Temple by the Babylonians.
Note that leading Jewish scholars over the years have come to the conclusion that the 1st Temple was constructed so that the Ark would have 2 possible resting places - first, the public one in the Holy of Holies (which was not really public, as only the High Priest would see it - covered - and only on Yom Kippur, the Day of Atonement) and, second, in a hidden location in deep maze-like vaults designed specifically for it by Solomon. Why? Because the consecration/validity of the Temple was dependent upon the presence of God, which was only manifest between the cherubim (cherubs) on top of the Ark. No Ark, no valid Temple. This means the the Ark HAD to be present - even though not in the Holy of Holies - during the time of the 2nd Temple. There is simply no way that the prophets of the time would have allowed the 2nd Temple to have even been started, let alone consecrated and used, without the Ark being there.
Here is my speculation:
The Babylonians, who destroyed the 1st Temple and carried off what treasures they found, did not ever claim to have the Ark. The Romans, who destroyed the 2nd Temple and carried off what treasures they found, did not ever claim to have the Ark, nor does any memorial of theirs (specifically, the Arch of Titus) show anything like the Ark. So, neither empire got hold of it...which strongly suggests that the only other possibility (besides it being hidden under Temple Mount, that is) is the Ethiopian story that Menelik, one of Solomon’s sons, carried it back to Sheba. That is pretty unlikely - does anyone seriously think that someone like Solomon - who was very smart, very rich and powerful, and deemed by God Himself to be pious enough to build the Temple - would just let something like the Ark go? I don’t think so - it wasn’t exactly an easy object to handle, having killed people who inadvertently touched it. You didn’t just go and grab it (from a guarded place, no less), toss it in a wagon and hightail it out of there - you had to exercise extreme care (and, if it was so holy, you would WANT to do so, even without the risk of being killed for touching it).
So...no to the Babylonians, no to the Romans and almost certainly no to the Ethiopians, leaving only the Jews. Add to the weight of the evidence that there was an extensive amount of prophecy available to lots of people (and certainly to all of the Biblical prophets of the era), so they would have KNOWN that it needed to be hidden to be kept safe. I’m going with that conclusion until someone proves otherwise.
last i heard the Copts had it...They had better leave it alone.